On the additive structure of algebraic valuations of polynomial semirings II

For $\alpha \in \mathbb{C}$, let $\mathbb{N}_0[\alpha]$ be the subsemiring of~$\mathbb{C}$ obtained as a homomorphic image of the $\alpha$-evaluation map $\mathbb{N}_0[x] \to \mathbb{C}$ defined as $p(x) \mapsto p(\alpha)$ for each polynomial $p(x) \in \mathbb{N}_0[x]$. Fundamental arithmetic and atomic aspects of the additive structure of $\mathbb{N}_0[\alpha]$ were first studied by the second author and Correa-Morris (2022). In this paper, we continue the investigation, now from the valuation-theoretic perspective. We show that for any algebraic number $\alpha$, the additive monoid of $\mathbb{N}_0[\alpha]$ contains no additive irreducibles if and only if it is isomorphic to the direct product of finitely many isomorphic valuation monoids (monoids whose principal ideals form a chain under inclusion). For any algebraic number $\alpha \in (0,1)$, these valuation monoids are precisely those where $\alpha^{-1}$ is a Perron number having no positive conjugates other than itself. In addition, we offer a description of the algebraic parameters $\alpha$ for which the additive structure of $\mathbb{N}_0[\alpha]$ is a valuation monoid. Finally, we argue that the subset of $(0,1)$ consisting of all algebraic parameters $\alpha$ such that the additive structure of $\mathbb{N}_0[\alpha]$ is a valuation monoid is dense in $(0,1)$.